Here we deal with the numerical solution of optimum design problems using computers. Of the three numerical methods for solving elliptic partial differential equations, the finite element method (FEM) is the obvious one to choose to use when the domains are the unknowns. We see that the FEM yields much simpler gradients than either the finite difference method or the boundary element method; these two methods are presented in Chapter 8. The FEM is presented first for a Neumann problem. Two other cases, a Dirichlet problem and a transmission problem, are treated.